Finite-temperature Field Theory - Theoretical Physics (TIFR)

Finite-temperature Field Theory 

Aleksi Vuorinen 

CERN 

Initial Conditions in Heavy Ion Collisions 

Goa, India, September 2008 

logo 

Aleksi Vuorinen, CERN 

Finite-temperature Field Theory

Basics of Statistical Physics and Thermodynamics 

Scalar fields at finite temperature 

Interacting scalar fields 

Why is finite T /µ physics (≈ QCD) interesting? 

High energy physics applications 

◮ Heavy ion experiments ⇒ Need 

quantitive understanding of 

non-Abelian plasmas at 

◮ High T and small/moderate µ 

◮ Moderately large couplings 

◮ In and (especially) out of 

equilibrium 

◮ Early universe thermodynamics 

◮ Signatures of phase transitions 

◮ EW baryogenesis 

◮ Cosmic relics 

◮ Neutron star cores 

logo 






In these lectures we will, however, only cover the very basics of 

finite-T formalism, restricting ourselves to 

◮ Equilibrium physics — imaginary-time formalism 

◮ Purely perturbative tools 

For other approaches/setups, see in particular Guy Moore’s 

lectures... 

logo 






Plan of the lectures 

◮ Lecture 1: Basic tools in finite-temperature field theory 

◮ Reminder of statistical physics 

◮ Scalar field theories and equilibrium thermodynamics 

◮ Interacting scalar fields at T ≠ 0: λϕ 4 theory 

◮ Lecture 2: Gauge field theories and linear response 

◮ More tools for finite-temperature calculations 

◮ Gauge symmetry, QED and QCD 

◮ Linear response theory 

◮ Lecture 3: Special applications in QCD 

◮ Phases of hot and dense QCD 

◮ The IR problem 

◮ Dimensional reduction and high-T effective theories 

logo 






Some useful reading 

Material used in preparing these lectures: 

◮ J. Kapusta, Finite-temperature field theory 

◮ M. Le Bellac, Thermal field theory 

◮ A. Rebhan, Thermal gauge field theories, hep-ph/0105183 

◮ D. Rischke, Quark-gluon plasma in equilibrium, 

nucl-th/0305030 

◮ ... 

logo 






Outline 


Statistical QM 

Simple examples 


Partition function for scalar field theory 

Non-interacting examples 


Feynman rules at finite temperature 

Thermodynamics of ϕ 4 theory 

logo 








Outline 










logo 








Grand canonical ensemble 

◮ Assume system that is free to exchange both energy and 

particles with a reservoir 

◮ ”Grand canonical ensemble” 

◮ System described by temperature T and chemical 

potentials µ i 

◮ Basic operator statistical density matrix ˆρ 

[ 

] 

ˆρ ≡ Z −1 exp −β(Ĥ − µ ˆN i 

} {{ } i ) , 

≡ ¯H 

〈Â〉 = Tr[ˆρÂ] 

◮ 

β = 1/T and µ Lagrangian multipliers ensuring 

conservation of E and N i 

logo 








Grand canonical ensemble 

◮ Assume system that is free to exchange both energy and 

particles with a reservoir 

◮ ”Grand canonical ensemble” 

◮ System described by temperature T and chemical 

potentials µ i 

◮ Basic operator statistical density matrix ˆρ 

[ 

] 

ˆρ ≡ Z −1 exp −β(Ĥ − µ ˆN i 

} {{ } i ) , 

≡ ¯H 

〈Â〉 = Tr[ˆρÂ] 

◮ 

β = 1/T and µ Lagrangian multipliers ensuring 

conservation of E and N i 

logo 








◮ Most important quantity in equilibrium thermodynamics: 

The partition function Z 

Z (V , T , µ i ) = Tr e −β ¯H = ∑ i 

〈i|e −β ¯H|i〉 = ∑ e −βE i 

◮ Several thermodynamic quantities available through 

derivatives of Z (due to above relation for expectation 

values) 

p = −Ω/V = 1 

βV ln Z , 

N i = 1 V 〈 ˆN i 〉 = ∂P 

∂µ i 

ε = 1 V 〈Ĥ〉 = − 1 V 

∂ln Z 

∂β 

logo 








◮ Most important quantity in equilibrium thermodynamics: 

The partition function Z 

Z (V , T , µ i ) = Tr e −β ¯H = ∑ i 

〈i|e −β ¯H|i〉 = ∑ e −βE i 

◮ Several thermodynamic quantities available through 

derivatives of Z (due to above relation for expectation 

values) 

p = −Ω/V = 1 

βV ln Z , 

N i = 1 V 〈 ˆN i 〉 = ∂P 

∂µ i 

ε = 1 V 〈Ĥ〉 = − 1 V 

∂ln Z 

∂β 

logo 








s = S/V = 1 V 〈−lnˆρ〉 = 1 V ln Z + β V 〈Ĥ − µ i ˆN i 〉 

= ∂P 

∂T = β(P + ε − µ iN i ) 

◮ Finally obtain energy through other quantities as 

E = −PV + TS + µ i N i 

◮ Important to note: Thermodynamic pressure = 

hydrodynamics pressure = 1 3 〈T ii 〉 

logo 









= ∂P 

∂T = β(P + ε − µ iN i ) 


E = −PV + TS + µ i N i 



logo 









= ∂P 

∂T = β(P + ε − µ iN i ) 


E = −PV + TS + µ i N i 



logo 








QM harmonic oscillator 

Use energy basis to obtain (in canonical ensemble): 

logo 








One degree of freedom 

◮ Consider one state system with E = ω, and ignore zero 

point energy 

Ĥ | n〉 = ω ˆN | n〉 = n ω | n〉 

◮ For bosons/fermions with chemical potential µ, obtain 

∞∑ 

Z b = Tr e −β(ω−µ) ˆN = 〈n | e −β(ω−µ) ˆN 1 

| n〉 = 

1 − e , 

−β(ω−µ) 

Z f = Tr e −β(ω−µ) ˆN = 

n=1 

1∑ 

〈n | e −β(ω−µ) ˆN | n〉 = 1 + e −β(ω−µ) 

n=1 

◮ Taking derivatives... 

N b = 

1 

e β(ω−µ) − 1 , E b = ωN b 

N f = 

1 

e β(ω−µ) + 1 , E f = ωN f 

logo 










point energy 

Ĥ | n〉 = ω ˆN | n〉 = n ω | n〉 


∞∑ 


| n〉 = 

1 − e , 

−β(ω−µ) 


n=1 

1∑ 

〈n | e −β(ω−µ) ˆN | n〉 = 1 + e −β(ω−µ) 

n=1 


N b = 

1 

e β(ω−µ) − 1 , E b = ωN b 

N f = 

1 

e β(ω−µ) + 1 , E f = ωN f 

logo 










point energy 

Ĥ | n〉 = ω ˆN | n〉 = n ω | n〉 


∞∑ 


| n〉 = 

1 − e , 

−β(ω−µ) 


n=1 

1∑ 

〈n | e −β(ω−µ) ˆN | n〉 = 1 + e −β(ω−µ) 

n=1 


N b = 

1 

e β(ω−µ) − 1 , E b = ωN b 

N f = 

1 

e β(ω−µ) + 1 , E f = ωN f 

logo 








Outline 










logo 








Partition function 

◮ Consider real scalar field theory with no chemical 

potentials 

∫ ∫ 

S = dt d 3 xL, 

L = 1 2 

{ 

(∂ t φ) 2 − (∂ i φ) 2 } − V (φ) 

◮ To obtain equilibrium thermodynamics, want to compute 

partition function 

∫ 

Z = Dφ〈φ | e −βĤ | φ〉 

logo 








Partition function 

◮ Consider real scalar field theory with no chemical 

potentials 

∫ ∫ 

S = dt d 3 xL, 

L = 1 2 

{ 

(∂ t φ) 2 − (∂ i φ) 2 } − V (φ) 

◮ To obtain equilibrium thermodynamics, want to compute 

partition function 

∫ 

Z = Dφ〈φ | e −βĤ | φ〉 

logo 








◮ Recall path integral for transition amplitudes in QFT: 

〈φ b (x) | e −iĤ(t 1−t 0 ) | φ a (x)〉 

∫ [ ∫ t1 

∫ ] 

= N Dϕ exp i dt d 3 x L 

t 0 

ϕ(t 0 ,x)=φa(x) 

ϕ(t 1 ,x)=φ b (x) 

◮ For thermodynamic purposes, set t 0 = t 1 , t → −iτ: 

∫ β−it0 ∫ 

iS → − dτ d d xL E , 

L E = 1 2 

−it 0 

{ 

(∂ t ϕ) 2 + (∂ i ϕ) 2 } + V (ϕ) 

◮ Finally, taking the trace... 

∫ 

[ 

Z = N Dϕ exp 

periodic 

∫ β−it0 ∫ 

− dτ 

−it 0 

d d xL E 

] 

logo 









〈φ b (x) | e −iĤ(t 1−t 0 ) | φ a (x)〉 

∫ [ ∫ t1 

∫ ] 


t 0 

ϕ(t 0 ,x)=φa(x) 

ϕ(t 1 ,x)=φ b (x) 


∫ β−it0 ∫ 


L E = 1 2 

−it 0 

{ 

(∂ t ϕ) 2 + (∂ i ϕ) 2 } + V (ϕ) 


∫ 

[ 

Z = N Dϕ exp 

periodic 

∫ β−it0 ∫ 

− dτ 

−it 0 

d d xL E 

] 

logo 









〈φ b (x) | e −iĤ(t 1−t 0 ) | φ a (x)〉 

∫ [ ∫ t1 

∫ ] 


t 0 

ϕ(t 0 ,x)=φa(x) 

ϕ(t 1 ,x)=φ b (x) 


∫ β−it0 ∫ 


L E = 1 2 

−it 0 

{ 

(∂ t ϕ) 2 + (∂ i ϕ) 2 } + V (ϕ) 


∫ 

[ 

Z = N Dϕ exp 

periodic 

∫ β−it0 ∫ 

− dτ 

−it 0 

d d xL E 

] 

logo 








A few things to note: 

◮ Periodicity of fields: Integrate over all ϕ satisfying 

ϕ(−it 0 , x) = ϕ(β − it 0 , x) 

◮ Simple consequence of taking the trace 

◮ Compactness of time direction 

◮ For equilibrium thermodynamics, may set t 0 = 0 

◮ Real Gaussian path integral 

◮ No issues of existence/convergence 

◮ Inclusion of finite chemical potentials straightforward 

(coming up in detail...) 

logo 








Side remark: Direct proof of periodicity 

◮ Define thermal (time-ordered) Green’s function 

[ 

G B (x, y; τ, 0) = Tr 

{ˆρT τ ˆφ(x, τ) ˆφ(y, 0) ]} 

◮ Using cyclic property of the trace... 

{ 

G B (x, y; τ, 0) = Z −1 Tr e −βĤe } 

βĤ ˆφ(y, 0)e −βĤ ˆφ(x, τ) 

{ 

} 

= Z −1 Tr e −βĤ ˆφ(y, β) ˆφ(x, τ) 

= Z −1 Tr 

{e −βĤT [ ] 

τ ˆφ(x, τ) ˆφ(y, } 

β) 

= G B (x, y; τ, β) 

logo 








Massive scalar fields 

◮ Specialize now to V (ϕ) = 1 2 m2 ϕ 2 

◮ 

One free bosonic dof 

◮ Following rules of Gaussian integration 

{ 

} 

( 

) −1/2 

ln Z = ln N det (−∂τ 2 − ∂i 2 + m 2 ) 

= − 1 2 Tr ln (−∂2 τ − ∂i 2 + m 2 ) + const. 

◮ How to take a trace of an operator containing derivatives? 

◮ 

Go to momentum space! 

logo 










◮ 



{ 

} 

( 

) −1/2 

ln Z = ln N det (−∂τ 2 − ∂i 2 + m 2 ) 

= − 1 2 Tr ln (−∂2 τ − ∂i 2 + m 2 ) + const. 


◮ 


logo 










◮ 



{ 

} 

( 

) −1/2 

ln Z = ln N det (−∂τ 2 − ∂i 2 + m 2 ) 

= − 1 2 Tr ln (−∂2 τ − ∂i 2 + m 2 ) + const. 


◮ 


logo 










◮ 



{ 

} 

( 

) −1/2 

ln Z = ln N det (−∂τ 2 − ∂i 2 + m 2 ) 

= − 1 2 Tr ln (−∂2 τ − ∂i 2 + m 2 ) + const. 


◮ 


logo 








◮ Expand fields in terms of their Fourier modes 

◮ 

ϕ(τ, x) = T ∑ ∫ 

d 3 p 

ei(p·x+ωnτ) 

(2π) 3 ϕ n (p), 

n 

ω n = 2nπT , n ∈ Z 

Fourier series in τ direction due to compactness of 

temporal direction 

◮ Now, a short exercise produces (up to T -indep. constant) 

ln Z = − V ∑ 

∫ 

d 3 ( 

p 

2 (2π) 3 ln n 2 + β2 (p 2 + m 2 ) 

) 

(2π) 2 n 

∫ [√ 

d 3 p p 

= −V 

2 + m 2 

(2π) 3 + ln 

(1 − e −β√ p 2 +m 2) ] 

2T 

◮ Reminiscent of point particle result for bosons! 

logo 









◮ 

ϕ(τ, x) = T ∑ ∫ 

d 3 p 


(2π) 3 ϕ n (p), 

n 

ω n = 2nπT , n ∈ Z 




ln Z = − V ∑ 

∫ 

d 3 ( 

p 

2 (2π) 3 ln n 2 + β2 (p 2 + m 2 ) 

) 

(2π) 2 n 

∫ [√ 

d 3 p p 

= −V 

2 + m 2 

(2π) 3 + ln 

(1 − e −β√ p 2 +m 2) ] 

2T 


logo 









◮ 

ϕ(τ, x) = T ∑ ∫ 

d 3 p 


(2π) 3 ϕ n (p), 

n 

ω n = 2nπT , n ∈ Z 




ln Z = − V ∑ 

∫ 

d 3 ( 

p 

2 (2π) 3 ln n 2 + β2 (p 2 + m 2 ) 

) 

(2π) 2 n 

∫ [√ 

d 3 p p 

= −V 

2 + m 2 

(2π) 3 + ln 

(1 − e −β√ p 2 +m 2) ] 

2T 


logo 








Brief note on fermions 

◮ For free Dirac fermions, going from QFT transition 

amplitude to finite-T partition function proceeds as above 

∫ 

[ ∫ β ∫ 

Z = D ¯ψDψ exp − dτ d 3 xL E 

], 

{ 

} 

L = ¯ψ γ 0 ∂ τ − iγ i ∂ i + m ψ 

0 

◮ What about periodicity? 

◮ Spinor fields satisfy canonical anticommutation relations 

and are represented by Grassmann variables 

◮ Let’s try to repeat our proof of periodicity for bosonic 

fields... 

logo 











∫ 

[ ∫ β ∫ 


], 

{ 

} 

L = ¯ψ γ 0 ∂ τ − iγ i ∂ i + m ψ 

0 





fields... 

logo 











∫ 

[ ∫ β ∫ 


], 

{ 

} 

L = ¯ψ γ 0 ∂ τ − iγ i ∂ i + m ψ 

0 





fields... 

logo 








◮ Define again thermal Green’s function 

[ ] 

G F (x, y; τ, 0) = Tr 

{ˆρT τ ˆψ(x, τ) ˆψ(y, } 

0) 

◮ 

This time field operators anticommute! 


{ 

G F (x, y; τ, 0) = Z −1 Tr e −βĤe } 

βĤ ˆψ(y, 0)e −βĤ ˆψ(x, τ) 

} 

= Z −1 Tr 

{e −βĤ ˆψ(y, β) ˆψ(x, τ) 

= −Z −1 Tr 

{e −βĤT [ 

τ ˆψ(x, τ) ˆψ(y, β) ]} 

= −G F (x, y; τ, β) 

logo 








◮ Define again thermal Green’s function 

[ ] 

G F (x, y; τ, 0) = Tr 

{ˆρT τ ˆψ(x, τ) ˆψ(y, } 

0) 

◮ 

This time field operators anticommute! 


{ 

G F (x, y; τ, 0) = Z −1 Tr e −βĤe } 

βĤ ˆψ(y, 0)e −βĤ ˆψ(x, τ) 

} 

= Z −1 Tr 

{e −βĤ ˆψ(y, β) ˆψ(x, τ) 

= −Z −1 Tr 

{e −βĤT [ 

τ ˆψ(x, τ) ˆψ(y, β) ]} 

= −G F (x, y; τ, β) 

logo 








◮ Conclusion: Fermion fields antiperiodic in time inside the 

path integral! 

ψ(β, x) = −ψ(0, x) 

⇒ 

ψ(τ, x) = T ∑ ∫ 

d 3 p 


(2π) 3 ψ n (p), 

n 

ω n = (2n + 1)πT , n ∈ Z 

logo 








Outline 










logo 








Scalar field theory with self-interactions 

◮ Consider a scalar field theory with interaction term 

proportional to small parameter λ 

◮ Action S = S0 + λS I 

◮ Now expand in coupling 

{ ∫ 

ln Z = ln N 

periodic 

Dϕ e −(S 0+λS I ) 

{ ∫ 

} 

∞∑ 

= ln N Dϕ e −S (−λS 

0 

I ) k 

periodic 

k! 

k=0 

{ 

∫ } 

∞∑ (−λ) k Dϕ e 

−S 0 

SI 

k = ln Z 0 + ln 1 + 

∫ 

k! Dϕ e −S 0 

k=1 

◮ Note underlying assumption here: [S0 , S I ] = 0 

} 

logo 








Scalar field theory with self-interactions 

◮ Consider a scalar field theory with interaction term 

proportional to small parameter λ 

◮ Action S = S0 + λS I 

◮ Now expand in coupling 

{ ∫ 

ln Z = ln N 

periodic 

Dϕ e −(S 0+λS I ) 

{ ∫ 

} 

∞∑ 

= ln N Dϕ e −S (−λS 

0 

I ) k 

periodic 

k! 

k=0 

{ 

∫ } 

∞∑ (−λ) k Dϕ e 

−S 0 

SI 

k = ln Z 0 + ln 1 + 

∫ 

k! Dϕ e −S 0 

k=1 

◮ Note underlying assumption here: [S0 , S I ] = 0 

} 

logo 








◮ Conclusion: In evaluating Z , end up computing expectation 

values of powers of S I within unperturbed theory 

〈S k I 〉 0 = 

∫ 

Dϕ e 

−S 0 

S k I 

∫ 

Dϕ e −S 0 

◮ No external legs: Partition function available through the 

computation of vacuum diagrams 

◮ As usual, powers of λ count loop order 

◮ Challenge: Derive Feynman rules at finite temperature 

logo 










〈S k I 〉 0 = 

∫ 

Dϕ e 

−S 0 

S k I 

∫ 

Dϕ e −S 0 





logo 










〈S k I 〉 0 = 

∫ 

Dϕ e 

−S 0 

S k I 

∫ 

Dϕ e −S 0 





logo 








Feynman rules for φ 4 theory 

◮ Assume massive scalar theory with L I = ϕ 4 

◮ 

Graphs composed of four-ϕ vertices and propagators 

1 

D 0 (ω n , p) = 

ωn 2 + p 2 + m 2 

◮ Feynman rules almost the same as at T = 0: 

◮ As usual, only need to compute connected diagrams 

◮ Same symmetry factors, vertex functions, etc. 

◮ Only changes due to discrete values of p 0 : 

◮ In integration measure: ∫ d 4 p 

→ T ∑ ∫ d 3 p 

(2π) 4 

n (2π) 3 

◮ In vertices: δ (4) (P 1 − P 2 ) → δ n1 ,n 2 

δ (3) (p 1 − p 2 ) 

◮ Question: How to evaluate the necessary sum-integrals? 

◮ Will be covered in more detail later... 

logo 










◮ 


1 

D 0 (ω n , p) = 

ωn 2 + p 2 + m 2 






→ T ∑ ∫ d 3 p 

(2π) 4 

n (2π) 3 


δ (3) (p 1 − p 2 ) 



logo 










◮ 


1 

D 0 (ω n , p) = 

ωn 2 + p 2 + m 2 






→ T ∑ ∫ d 3 p 

(2π) 4 

n (2π) 3 


δ (3) (p 1 − p 2 ) 



logo 










◮ 


1 

D 0 (ω n , p) = 

ωn 2 + p 2 + m 2 






→ T ∑ ∫ d 3 p 

(2π) 4 

n (2π) 3 


δ (3) (p 1 − p 2 ) 



logo 








Example: Pressure of massless ϕ 4 theory 

◮ To obtain pressure of ϕ 4 theory to 4 loops, must evaluate 

◮ Up to two loops: 

( ∫ ∑ 

p(T ) = p 0 (T ) − 3λ 

≡ 

( ∫ ∑ 

p 0 (T ) − 3λ 

P 

P 

) 

1 2 

ωn 2 + p 2 

) 

1 2 

P 2 

where the sum-integral reads... 

logo 








Example: Pressure of massless ϕ 4 theory 

◮ To obtain pressure of ϕ 4 theory to 4 loops, must evaluate 

◮ Up to two loops: 

( ∫ ∑ 

p(T ) = p 0 (T ) − 3λ 

≡ 

( ∫ ∑ 

p 0 (T ) − 3λ 

P 

P 

) 

1 2 

ωn 2 + p 2 

) 

1 2 

P 2 

where the sum-integral reads... 

logo 








∑ ∫ P 

1 

P 2 = Λ 2ε T 

= 

= 

∞∑ 

n=−∞ 

∫ 

2Γ(−1/2 + ε) 

(4π) 3/2−ε 

d 3−2ε p 

(2π) 3−2ε 1 

p 2 + (2nπT ) 2 

Λ 2ε T (2πT ) 1−2ε ∞ ∑ 

n=1 

n 1−2ε 

2Γ(−1/2 + ε) 

(4π) 3/2−ε Λ 2ε T (2πT ) 1−2ε ζ(−1 + 2ε) 

= T 2 

12 + O(ε) 

◮ Recalling the free theory result, finally obtain 

p(T ) = π2 T 4 ( 

1 − 15 ) 

90 8π 2 λ + O(λ 2 ) 

logo 








∑ ∫ P 

1 

P 2 = Λ 2ε T 

= 

= 

∞∑ 

n=−∞ 

∫ 

2Γ(−1/2 + ε) 

(4π) 3/2−ε 

d 3−2ε p 

(2π) 3−2ε 1 

p 2 + (2nπT ) 2 

Λ 2ε T (2πT ) 1−2ε ∞ ∑ 

n=1 

n 1−2ε 

2Γ(−1/2 + ε) 

(4π) 3/2−ε Λ 2ε T (2πT ) 1−2ε ζ(−1 + 2ε) 

= T 2 

12 + O(ε) 

◮ Recalling the free theory result, finally obtain 

p(T ) = π2 T 4 ( 

1 − 15 ) 

90 8π 2 λ + O(λ 2 ) 

logo

Finite-temperature Field Theory - Theoretical Physics (TIFR)

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?